Twin Parado Revisied Relaiviy and Asrophysics Lecure 19 Terry Herer Ouline Simulaneiy Again Sample Problem L- Twin Parado Revisied Time dilaion viewpoin Lengh conracion viewpoin Parado & why i s no! Problem L-13, page 117 (due Friday/Monday) Will hand back on Monday if you hand i in on Friday A90-19 Twin Parado A90-19 1
Revisiing Simulaneiy and Paradoes We will look one more ime a Simulaneiy Time dilaion / win parado As we showed previously If wo evens occur in our lab frame separaed by disance and ime, we can use he Lorenz ransformaion o find a rocke frame in which Evens occur a same locaion (for < ) Evens occur a same ime (for > ) A90-19 Twin Parado 3 Timelike Inervals In lab frame we see wo evens (even A and even ) separaed by a disance (= A ) and ime (= A ) Can we now find a rocke frame in which he evens occur a he same locaion? Le he coordinaes for even A be zero in boh frames. Then we have A = A = 0 = known = 0 A = A = 0 = known = unknown We wan he speed v of he rocke such ha = 0. We use he Lorenz ransformaion equaions v v = 0 v Again v is he required speed of he rocke so ha = 0. Noe ha since v < 1, we mus have < Thus - > 0 (he spaceime inerval is imelike) A90-19 Twin Parado 4 A90-19
Evens a he same locaion Evens occur a same locaion v = 1 0 = 0 line, v = b / b When b < b (separaion < dela ime) for wo evens, we can always choose a frame in which he evens happens a he same locaion. 0 A90-19 Twin Parado 5 Spacelike Inervals In lab frame we see wo evens (even A and even ) separaed by a disance (= A ) and ime (= A ) Can we find a rocke frame in which he evens are simulaneous? Le he coordinaes for even A be zero in boh frames. Then we have A = A = 0 = known = unknown A = A = 0 = known = 0 We wan he speed v of he rocke such ha = 0. We use he Lorenz ransformaion equaions v v = 0 v v is he required speed of he rocke so ha = 0. Noe ha since v < 1, we mus have > Thus - < 0 (he spaceime inerval is spacelike) A90-19 Twin Parado 6 A90-19 3
Evens a he same ime v = 1 When b > b (separaion > dela ime) for wo evens, we can always choose a frame in which he evens happens a he same ime. = 0 line, v = b / b Evens occur a same ime A90-19 Twin Parado 7 Simulaneiy Revisied: Sample Prob. L- Problem Seup Julius Caesar was murdered approimae 000 years ago. Is here a way using he laws of physics o save his life? Le Caesar s deah be he reference even wih coordinaes, o = 0, o = 0 Even A is he presen ime which is A = 0 lyr, A ~ 000 yr Simulaneous wih even A, he Sarship Enerprise ses off a fire cracker in he Andromeda galay Even is he firecracker a = 10 6 lyr, = 000 yr. Le o = 0, o = 0 for he Enerprise (Caesar s murder is he reference even) A90-19 Twin Parado 8 A90-19 4
Sample Problem L- (con d) How fas mus he Enerprise be going in he Earh frame in order ha Caesar s murder is happening NOW, ha is, = 0? In lab frame we see wo evens (even A and even ) separaed by a disance (= A ) and ime (= A ) Can we find an Enerprise frame in which he evens are simulaneous? Le he coordinaes for even A be zero in boh frames. Then we have A = A = 0 = 10 6 yr = unknown A = A = 0 = 10 3 yr = 0 We wan he speed v of he rocke such ha = 0. We use he Lorenz ransformaion equaions v v = 0 v 3 10 yr 6 10 yr Thus v = 10-3 = 0.001 we have = 0, ha is, Caesar s murder happens NOW in he Enerprise frame, A90-19 Twin Parado 9 Sample Problem L- (con d) Draw a spaceime diagram which conains: Even O (Caesar s deah), Even A (you here), even (firecracker in Andromeda), your line of NOW simulaneiy, he posiion of he Enerprise, he worldline of he Enerprise, and he Enerprise NOW line of simulaneiy. You are viewing his Enerprise ses off firecracker in Andromeda Segmen of Enerprise worldline ime A Caesar s deah Earh line of simulaneiy Enerprise line of simulaneiy = v ( = 0 line) O space A90-19 Twin Parado 10 A90-19 5
Sample Problem L- (con d) In he Enerprise frame wha are he and coordinaes of he firecracker. Noe ha we choose he aive velociy of he Enerprise so ha = 0, which gives v = 10-3. Compue and use he inverse Lorenz ransformaion. v Coordinae is no much differen in he wo frames because he velociy is small 1 1 v v 6 v 10 ~ 1 1 3 6 3 10 10 10 0 yr 6 3 3 10 10 10 6 6 1 10 10 6 6 6 1 10 1 10 10 6 1 6 1 10 10 10 1.99999910 6 lyr A90-19 Twin Parado 11 Sample Problem L- (con d) Can he Enerprise firecracker eplosion warn Caesar, hus changing he course of hisory? There eiss a frame (he res frame of he Enerprise) in which Caesar s deah and he firecracker eplosion occur a he same ime. In his frame a signal connecing he wo evens would have o ravel faser han ligh. Enerprise can warn Caesar. The spaial coordinae is hugely differen Evens separaed in his fashion are called spacelike Spacelike evens canno be causally linked, ha is, hey canno have a cause and effec aionship. A90-19 Twin Parado 1 A90-19 6
Twin Parado From: Lecure, slide 8, using spaceime inerval A sarship leaves Earh (Even 1) and ravel a 95% ligh speed, laer arriving a Proima Cenauri (Even ), which lies 4.3 ligh-years from Earh. 1. Wha are he space and ime separaion beween he wo evens as measured in he Earh frame, in years?. Wha are he space and ime separaions in he frame of he sarship? Par 1: The disance separaion is 4.3 ligh-years, as saed in he problem. The ime separaion is: Par : oh evens occur a he posiion of he sarship => space separaion is 0 m in sarship frame. Once more, we use invariance of spaceime inerval o ge ime separaion in he proon frame: d d ship earh 4.3 lyr 0.95 (lyr/yr) A90-19 Twin Parado 13 ship 4.56 yr 4.3 yr 0.48 18.49yr 1.99 yr 4.56 years ship earh ship earh 1.41 years This is he famous win parado ime goes by more slowly on he sarship han Earh. The Twin Parado - revisied One win ravels away from he Earh a high speed for a long ime. Reurning o Earh she finds herself much younger han her broher! How? oh wins hink each oher s clock slows down, so wha is going on? Suppose one win is on a rocke ship raveling a v = 0.95 o a sar 4.3 lyr away. She sees he disance conraced o: L 4.3 1 0.95 lyr 1.34 lyr She compues her ime o ge here as 1.34 lyr/0.95 1.41 yr No warp drive The win on Earh sees he disance as sill 4.3 lyr, so he compues her ime o go ou as: 4.3 lyr/0.95 4.56 yr Analyze using Lorenz conracion So he ages 4.53 years, while his win siser ages only 1.41 years. A90-19 Twin Parado 14 A90-19 7
Relaiviy of Simulaneiy (chp. 4.9) How do we eplain he win parado oh wins see each oher aging less bu he one on he Earh ages more han he one in he sarship We will consider a round rip o Proima Cenauri Le us se up a se of observing (lookou) saions ha are associaed wih he Earh and wih he rocke ship We wan o see wha hey measure? Noe ha all hree frames are NOT he same The person in he rocke has o urn around o reurn! Thus he ougoing and reurn frames are compleely differen Thus we can hink of hese as wo differen rockes reurn-rocke lookou saions ougoing-rocke lookou saions Earh Earh lookou saions Proima Cenauri A90-19 Twin Parado 15 Ougoing Rocke So i akes 1.41 years by ougoing-rocke ime o reach Proima Cenauri (PC). Wha ime does he ougoing rocke lookou read on he Earh clock as she passes i (when he rocke reaches PC? Time dialaion mus be he same => 1.41(1.41/4.53) = 0.44 years! Noe wha we have This is wha he Earh clock reads a he same ime as he ougoing rocke arrives a PC as measured in he ougoing rocke frame. u a he same ime as he ougoing rocke arrives a PC, he Earh clock reads 4.53 years as measured in he Earhbound frame. Observers in differen reference frames do no agree on wha evens occur a he same ime when hese evens occur far apar along he line of moion. Called Relaiviy of Simulaneiy A90-19 Twin Parado 16 A90-19 8
Reurn Rocke When reurn rocke arrives a Earh he clocks read Rocke clock =.8 years, Earh clock = 9.06 years The readings are occurring a he same locaion so we don need o worry abou he aiviy of simulaneiy According o reurn rocke frame observaions he Earh clock again records an elapsed ime of 1.41(1.41/4.53) = 0.44 years on he reurn rip. Therefore a he urn around in he reurn rocke frame he Earh clock read 9.06 0.44 = 8.6 years However, in he ougoing rocke frame he Earh clock read 0.44 years when he rocked reached PC Which is righ? boh are! The wo observaions are from differen frames The jump is he resul of he raveler changing frames a Proima Cenauri. A90-19 Twin Parado 17 Parado Summary The eperiences of he rocke raveler and he Earh dweller are no symmeric. The raveler has o change direcions o reurn (or sop) => acceleraion Thus he raveler changes reference frames All frames are consisen and non-paradoical Earh-clock reading observed by Even Time measured in Earh-linked frame Time measured by raveler ougoing-rocke lookou saion passing Earh reurn-rocke lookou saion passing Earh Depar Earh 0 yr 0 yr 0 yr Arrive P. Cen 4.53 yr 1.41 yr 0.44 yr Depar P. Cen 4.53 yr 1.41 yr 0.44 yr 8.6 yr Arrive Earh 9.06 yr.8 yr 9.06 yr A90-19 Twin Parado 18 A90-19 9
Travel Time Twin Parado In Euclidean geomery he shores pah lengh beween wo poins is by a ravel moving in a sraigh line Or he raveler who changes direcion he leas! In Spaceime, he greaes aging beween wo evens is eperience by he raveler who does no change direcion. d d d d d The minus sign means ha d has o be larger han in a frame which moves beween evens (d = 0). See Chaper 4.6 of Spaceime Physics A90-19 Twin Parado 19 A90-19 10